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Part of 2020 Mediterranean Mathematics Olympiad

Problems(1)

at least n^2 integers written in the form x+yz with x,y,z\in S

Source: Mediterranean Mathematical Olympiad 2020 P2 MMC

9/23/2020
Let SS be a set of n≥2n\ge2 positive integers. Prove that there exist at least n2n^2 integers that can be written in the form x+yzx+yz with x,y,z∈Sx,y,z\in S.
Proposed by Gerhard Woeginger, Austria
number theorycombinatorics